General Certificate of Secondary Education (GCSE) Mathematics Flashcards

24 sample cards from the Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. How do you share an amount in a given ratio, e.g. £$60$ in the ratio $2:3$?

   Add the parts: $2+3=5$. One part $=\frac{60}{5}=£12$. So the shares are $2\times12=£24$ and $3\times12=£36$.

2. What is the difference between direct and inverse proportion?

   Direct proportion: $y=kx$ (as one increases the other increases at a constant ratio). Inverse proportion: $y=\frac{k}{x}$ (as one increases the other decreases). $k$ is the constant of proportionality.

3. State the formulas for speed, density and pressure (compound measures).

   $\text{speed}=\frac{\text{distance}}{\text{time}}$, $\quad \text{density}=\frac{\text{mass}}{\text{volume}}$, $\quad \text{pressure}=\frac{\text{force}}{\text{area}}$.

4. What is the compound interest formula?

   $$A=P\left(1+\frac{r}{100}\right)^{n}$$ where $P$ is the principal, $r$ the percentage rate per period, and $n$ the number of periods. For decay, use $\left(1-\frac{r}{100}\right)^{n}$.

5. State the sum of interior angles of a polygon and the size of each exterior angle of a regular polygon.

   Sum of interior angles $=(n-2)\times180^{\circ}$. Each exterior angle of a regular $n$-gon $=\frac{360^{\circ}}{n}$, and exterior angles always sum to $360^{\circ}$.

6. State the angle facts: angles on a straight line, around a point, and in a triangle.

   Angles on a straight line sum to $180^{\circ}$; angles around a point sum to $360^{\circ}$; angles in a triangle sum to $180^{\circ}$.

7. Give the area formulas for a triangle, parallelogram, trapezium and circle.

   Triangle: $\frac{1}{2}bh$. Parallelogram: $bh$. Trapezium: $\frac{1}{2}(a+b)h$. Circle: $\pi r^{2}$.

8. Give the circumference of a circle and the arc length / sector area of a sector with angle $\theta$.

   Circumference $=2\pi r=\pi d$. Arc length $=\frac{\theta}{360}\times2\pi r$. Sector area $=\frac{\theta}{360}\times\pi r^{2}$.

9. State the volume and surface area of a sphere, and the volume of a cone.

   Sphere volume $=\frac{4}{3}\pi r^{3}$, surface area $=4\pi r^{2}$. Cone volume $=\frac{1}{3}\pi r^{2}h$, curved surface area $=\pi r l$.

10. State Pythagoras' theorem.

    In a right-angled triangle, $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse (the side opposite the right angle).

11. State the three basic trigonometric ratios (SOHCAHTOA).

    $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\quad\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\quad\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

12. Give the exact values of $\sin$, $\cos$ and $\tan$ for $30^{\circ}$, $45^{\circ}$ and $60^{\circ}$.

    $\sin30^{\circ}=\frac{1}{2}$, $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, $\tan30^{\circ}=\frac{1}{\sqrt{3}}$; $\sin45^{\circ}=\cos45^{\circ}=\frac{\sqrt{2}}{2}$, $\tan45^{\circ}=1$; $\sin60^{\circ}=\frac{\sqrt{3}}{2}$, $\cos60^{\circ}=\frac{1}{2}$, $\tan60^{\circ}=\sqrt{3}$.

13. State the sine rule and the cosine rule for any triangle.

    Sine rule: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Cosine rule: $a^{2}=b^{2}+c^{2}-2bc\cos A$. Area $=\frac{1}{2}ab\sin C$.

14. Name the four types of transformation and what defines each.

    Translation (described by a column vector), reflection (in a mirror line), rotation (about a centre, with angle and direction), and enlargement (by a scale factor from a centre). The first three preserve size and shape; enlargement changes size.

15. What does a negative or fractional enlargement scale factor do?

    A fractional scale factor (between $0$ and $1$) makes the shape smaller. A negative scale factor enlarges through the centre of enlargement, producing an image on the opposite side that is inverted.

16. How do you add column vectors and find the magnitude of a vector?

    Add component-wise: $\begin{pmatrix}a\\b\end{pmatrix}+\begin{pmatrix}c\\d\end{pmatrix}=\begin{pmatrix}a+c\\b+d\end{pmatrix}$. Magnitude $=\sqrt{a^{2}+b^{2}}$.

17. How do you construct the perpendicular bisector of a line segment $AB$?

    Place the compass at more than half of $AB$. Draw arcs centred on $A$ and on $B$ above and below the line; they intersect at two points. Draw the straight line through those intersections — it bisects $AB$ at right angles, and every point on it is equidistant from $A$ and $B$.

18. What is the locus of points a fixed distance from a point, and from a line?

    The locus a fixed distance from a point is a circle of that radius centred on the point. The locus a fixed distance from a straight line is a pair of parallel lines (and, around a line segment, semicircular ends forming a 'racetrack').

19. State the probability scale and the formula for the probability of an event.

    Probability runs from $0$ (impossible) to $1$ (certain). For equally likely outcomes, $P(\text{event})=\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$.

20. State the AND and OR rules for independent / mutually exclusive events.

    For independent events (AND): $P(A\text{ and }B)=P(A)\times P(B)$. For mutually exclusive events (OR): $P(A\text{ or }B)=P(A)+P(B)$. Also $P(\text{not }A)=1-P(A)$.

21. What is the difference between a primary and secondary data source, and between a sample and a population?

    Primary data is collected first-hand by the investigator; secondary data is taken from an existing source. The population is the whole group being studied; a sample is a representative subset used to make inferences about it.

22. How do you calculate the mean, median, mode and range of a data set?

    Mean $=\frac{\text{sum of values}}{\text{number of values}}$. Median is the middle value when ordered. Mode is the most frequent value. Range $=\text{largest}-\text{smallest}$.

23. What does a box plot show, and how is the interquartile range found?

    A box plot shows the minimum, lower quartile $Q_1$, median $Q_2$, upper quartile $Q_3$ and maximum. The interquartile range (IQR) $=Q_3-Q_1$ and measures the spread of the middle $50\%$ of the data.

24. How do you estimate the mean from a grouped frequency table?

    Use the midpoint $x$ of each class. Estimated mean $=\frac{\sum fx}{\sum f}$, the sum of (frequency $\times$ midpoint) divided by the total frequency.

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